Monte Carlo Study of Lattice Compact Quantum Electrodynamics with Fermionic Matter: The Parent State of Quantum Phases

The interplay between lattice gauge theories and fermionic matter accounts for fundamental physical phenomena ranging from the deconfinement of quarks in particle physics to quantum spin liquid with fractionalized anyons and emergent gauge structures in condensed matter physics. However, except for certain limits (for instance, a large number of flavors of matter fields), analytical methods can provide few concrete results. Here we show that the problem of compact $\mathrm{U}(1)$ lattice gauge theory coupled to fermionic matter in $(2+1)\mathrm{D}$ is possible to access via sign-problem-free quantum Monte Carlo simulations. One can hence map out the phase diagram as a function of fermion flavors and the strength of gauge fluctuations. By increasing the coupling constant of the gauge field, gauge confinement in the form of various spontaneous-symmetry-breaking phases such as the valence-bond solid (VBS) and Néel antiferromagnet emerge. Deconfined phases with algebraic spin and VBS correlation functions are also observed. Such deconfined phases are incarnations of exotic states of matter, i.e., the algebraic spin liquid, which is generally viewed as the parent state of various quantum phases. The phase transitions between the deconfined and confined phases, as well as that between the different confined phases provide various manifestations of deconfined quantum criticality. In particular, for four flavors $N_f=4$, our data suggest a continuous quantum phase transition between the VBS and Néel order. We also provide preliminary theoretical analysis for these quantum phase transitions.

Popular Summary

An exotic 2D state of matter known as a U(1) Dirac spin liquid could help condensed-matter physicists explain perplexing observations of many high-temperature superconductors and frustrated magnets. High-energy physicists are also interested in these spin liquids as they might be a key to understanding predictions from quantum electrodynamics of a “deconfined” state of matter, one where elementary particles split from a conventional description of particles or fields. And yet neither community knows whether this phase even exists. Here, we use computer simulations to provide, for the first time, concrete evidence of its existence.

The emergence of a U(1) Dirac spin liquid (or deconfined matter, in the parlance of high-energy physics) depends on two ingredients: a U(1) gauge field and a spinon. A spinon is a “fractional” spin of a fermion, whereas the U(1) gauge field mediates interactions between spinons. It is thought that a spin liquid arises when spinons and their gauge field are put on a 2D square lattice. To explore this possibility, we design a model by discretizing U(1) gauge fields into a space-time lattice that interacts with fermions.

Using quantum Monte Carlo methods, we simulate the model with various fermion species and find, in all cases, a smoking-gun signature of 2D U(1) deconfined matter. By tuning coupling constants in the model, we also observe continuous transitions between deconfined and confined states. One fermion species also exhibits a novel type of quantum critical point.

We hope that the richness of the phase diagram and firm existence of the U(1) deconfined state of matter will trigger a number of future investigations on the numerical and analytical fronts in the condensed-matter and high-energy communities.

Figure 1

(a) Phase diagram spanned by the Fermi flavors $N_f$ and the strength of gauge field fluctuations $J$ of the model shown in (b). $\mathrm{U}(1)\mathrm{D}$ stands for the $\mathrm{U}(1)$ deconfined phase where the fermions dynamically form a Dirac system. This phase corresponds to the algebraic spin liquid where all correlation functions show slow power-law decay. VBS stands for valence-bond-solid phase and AFM stands for the antiferromagnetic long-range ordered phase (Néel phase). (b) Sketch of the model of Eq. (1). The yellow circles represent the gauge field attached to each fermion hopping, and the yellow dashed lines stand for the flux term per plaquette. (c) The gauge-invariant propagator for fermions with a string of gauge fields attached.

Figure 2

(a) The antiferromagnetic correlation ratio through the $\mathrm{U}(1)\mathrm{D}$-to-AFM transition at $N_f=2$. Here, $\beta =4L$, $\mathrm{\Delta }\tau =0.2$. The crossing points are the transition points separating the deconfined phase and Néel phase. (b) The $1/L$ extrapolation of the crossings estimates the $\mathrm{U}(1)\mathrm{D}$-to-AFM transition point $J_c=1.6(2)$ for $N_f=2$.

Figure 3

Photon mass $m$ measured by the flux correlation $C(\tau )=⟨\theta ({\tau }_0)\theta ({\tau }_0+\tau )⟩\sim \exp (-m\tau )$ with $\theta (\tau )={\sum }_{\square }\sin [\mathrm{curl}\varphi (\tau )]$. Zero photon mass is observed in the $\mathrm{U}(1)\mathrm{D}$ phase; finite photon mass is observed in the confined phase. Here we plot the $N_f=2$ case.

Figure 4

The log-log plot of real-space decay of (a) spin correlation functions and (b) dimer correlation functions for $N_f=2$ in the $\mathrm{U}(1)\mathrm{D}$ phase (at $J=1.25<J_c$). The slope gives a good estimation of the scaling dimension of spin and dimer. Note that to avoid even-odd oscillation in the finite-size data, here only the distance $r=$ odd points are plotted in the $\mathrm{U}(1)\mathrm{D}$ phase. For other $N_f$ cases in the following, we adopt the same strategy.

Figure 5

The VBS correlation ratio through the $\mathrm{U}(1)\mathrm{D}$-to-VBS transition at $N_f=4$. Here, $\beta =3L$, $\mathrm{\Delta }\tau =0.2$. (b) The $1/L$ extrapolation of the crossings estimates the $\mathrm{U}(1)\mathrm{D}$-to-VBS transition point $J_{c1}=1.2(3)$ for $N_f=4$.

Figure 6

The log-log plot of real-space decay of (a) spin correlation functions and (b) dimer correlation functions for $N_f=4$ in the $\mathrm{U}(1)\mathrm{D}$ phase (at $J=1.00<J_c$). The slope gives a good estimation of the scaling dimension of the spin and dimer.

Figure 7

Dimension of spin (gren circles) and dimer (blue squares) in the $\mathrm{U}(1)\mathrm{D}$ phase as a function of $N_f$. The dashed red line corresponds to the $1/N_f$ perturbative calculation, $1+\eta =4-[64/(3{\pi }^2{N}_f)]$, taken from Ref. [7]. Note here $N_f$ corresponds to the number of four-component Dirac fermions.

Figure 8

(a) Antiferromagnetic correlation ratio $r_{\mathrm{AFM}}$ for $N_f=4$. Here, $\beta =3L$, $\mathrm{\Delta }\tau =0.2$. Insets show the $1/L$ extrapolation of the crossing point in $r_{\mathrm{AFM}}$ and $J_{c2}=18(3)$. (b) VBS correlation ratio $r_{\mathrm{VBS}}$ for $N_f=4$. The inset is the $1/L$ extrapolation of the crossing point in $r_{\mathrm{VBS}}$ and $J_{c2}=19(5)$. This is consistent with $J_{c2}$ obtained from $r_{\mathrm{AFM}}$ in (a). (c) AFM-VBS correlation ratio $r_{\mathrm{AFM}\text{−}\mathrm{VBS}}$ for $N_f=4$. The inset is the $1/L$ extrapolation of the crossing point in $r_{\mathrm{AFM}\text{−}\mathrm{VBS}}$ and $J_{c2}=17(4)$.

Figure 9

The VBS correlation ratio through the $\mathrm{U}(1)\mathrm{D}$-to-VBS transition at $N_f=6$. Here, $\beta =2L$, $\mathrm{\Delta }\tau =0.1$. (b) The $1/L$ extrapolation of the crossings estimates the $\mathrm{U}(1)\mathrm{D}$-to-VBS transition point at $J_c=1.9(3)$ for $N_f=6$.

Figure 10

The log-log plot of real-space decay of (a) spin correlation functions and (b) dimer correlation functions for $N_f=6$ in the $\mathrm{U}(1)\mathrm{D}$ phase (at $J=1.40<J_c$). The slope gives a good estimation of the scaling dimension of the spin and dimer.

Figure 11

The log-log plot of real-space decay of (a) spin correlation functions and (b) dimer correlation functions for $N_f=8$ in the $\mathrm{U}(1)\mathrm{D}$ phase (at $J=2.00<J_c$). The slope gives a good estimation of the scaling dimension of the spin and dimer.

Figure 12

The VBS correlation ratio through the $\mathrm{U}(1)\mathrm{D}$-to-VBS transition at $N_f=8$. Here, $\beta =2L$, $\mathrm{\Delta }\tau =0.1$. (b) The $1/L$ extrapolation of the crossings estimates the $\mathrm{U}(1)\mathrm{D}$-to-VBS transition point at $J_c=2.5(1)$ for $N_f=8$.

Figure 13

Flux energy per plaquette along the same $J$ path. There is no singularity around $J_c$, suggesting it is a continuous phase transition.

Figure 14

Uniform structure factor of ${\widehat{Q}}_i$ [defined in Eq. (e1)] for $N_f=2$. Note that $\beta =4L$ here. For all $J$’s, this uniform structure factor extrapolated to zero in thermodynamic limit. Thus, the constraint of ${\widehat{Q}}_i=0$ is dynamically enforced.

Figure 15

Local update acceptance ratio at $N_f=2$ with $L=6$. Note the acceptance ratio for other sizes is almost the same, not shown here.

Figure 16

Net flux sweep serials in the time-slice plane $\tau$ and ${\tau }^{\prime }$ at $N_f=2$ with $L=12$ (a) inside the $\mathrm{U}(1)\mathrm{D}$ phase and (b) inside the AFM phase. Here, $\tau =\mathrm{\Delta }\tau$ and ${\tau }^{\prime }=8\mathrm{\Delta }\tau$. The flux sweep serials are plotted in intervals of 20 sweeps, and an $L^2/2$ is added to shift it to be centred around zero.